Calculus II · Unit 4B · lesson
Radius and Interval of Convergence
Find the centered interval structure of power-series convergence and separate automatic interior behavior from endpoint testing.
Section overview
Power-series convergence and endpointsWhat this section is building
Find the centered interval structure of power-series convergence and separate automatic interior behavior from endpoint testing.
Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.
Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.
Including or excluding both endpoints from the radius calculation alone.
Learning objectives
find the radius of convergence and describe the automatic convergence and divergence regions.
Radius and Interval of Convergence
Power-series convergence has a rigid geometry
A power series has a remarkable all-or-nothing structure around its center. If it converges at a point some distance from the center, it converges absolutely at every closer point. If it diverges at one point, it diverges at every farther point. The result is a radius separating automatic convergence from automatic divergence.
The radius does not decide endpoint inclusion. At , the ratio or root calculation usually becomes exactly one and loses power. Each endpoint must be substituted into the original series and tested independently. The final interval may be open, closed, or half-open.
Convergence spreads inward from the center
Power-series convergence is organized by distance from the center . If the series converges at a point , then it converges absolutely at every point closer to . If it diverges at one point, it diverges at every point farther away. This produces a radius and an interval centered at .
The radius settles only the interior and exterior. At the boundary points and , the geometric domination used inside has ratio one and stops deciding the question. Each endpoint must be tested separately as an ordinary numerical series.
Read this graph as text
The radius organizes convergence by distance from the center. A number line centered at a shows an interior interval of absolute convergence, two separately marked endpoints, and exterior divergence regions. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in the radius organizes convergence by distance from the center; color is never the only cue.
Why it matters: A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.
A number line centered at a shows an interior interval of absolute convergence, two separately marked endpoints, and exterior divergence regions.
The radius organizes convergence by distance from the center. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.
Interval-of-convergence workflow
First find the center and radius by testing absolute convergence for a generic . Record the open interior interval. Then substitute the left endpoint into the original series and test it; repeat independently at the right endpoint. Only after both decisions should interval notation be written.
A closer point introduces a geometric factor
If converges, its terms are bounded. For , write
The bounded first factor times a geometric factor with ratio below one proves absolute convergence.
Read this graph as text
A power series converges on an interval centered at its expansion point. Inside the radius R, convergence is automatic; outside, divergence is automatic; each endpoint must be checked separately. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in a power series converges on an interval centered at its expansion point; color is never the only cue.
Why it matters: A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.
Inside the radius , convergence is automatic; outside, divergence is automatic; each endpoint must be checked separately.
A power series converges on an interval centered at its expansion point. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.
The open left endpoint and closed right endpoint are only an example. Endpoint inclusion depends on separate one-variable series tests.
Radius structure
For some , a power series centered at converges absolutely when and diverges when .
A basic radius
For
the geometric ratio is . Thus , so the radius is and the open convergence interval is . Endpoints still require tests.
Read the interval from the center and radius
Suppose a power series centered at has radius . It converges absolutely whenever
which is the open interval . It diverges for or . The points and remain unresolved until they are substituted into the original series.
Radius and interval are not the same object
The radius is the nonnegative number . The interval includes the center and records endpoint behavior, such as .
u4b-radius_and_interval_of_convergence-01Find the radius of convergence of .
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Require the geometric ratio magnitude to be below one.
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Find the radius of .
Find the center and radius of .
Explain what and mean.
Why does radius alone not settle endpoints?
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